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Properties

Label 1-235-235.207-r0-0-0
Degree $1$
Conductor $235$
Sign $-0.779 - 0.626i$
Analytic cond. $1.09133$
Root an. cond. $1.09133$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.631 − 0.775i)2-s + (−0.398 + 0.917i)3-s + (−0.203 − 0.979i)4-s + (0.460 + 0.887i)6-s + (−0.942 − 0.334i)7-s + (−0.887 − 0.460i)8-s + (−0.682 − 0.730i)9-s + (0.576 − 0.816i)11-s + (0.979 + 0.203i)12-s + (−0.997 − 0.0682i)13-s + (−0.854 + 0.519i)14-s + (−0.917 + 0.398i)16-s + (0.816 − 0.576i)17-s + (−0.997 + 0.0682i)18-s + (−0.990 − 0.136i)19-s + ⋯
L(s)  = 1  + (0.631 − 0.775i)2-s + (−0.398 + 0.917i)3-s + (−0.203 − 0.979i)4-s + (0.460 + 0.887i)6-s + (−0.942 − 0.334i)7-s + (−0.887 − 0.460i)8-s + (−0.682 − 0.730i)9-s + (0.576 − 0.816i)11-s + (0.979 + 0.203i)12-s + (−0.997 − 0.0682i)13-s + (−0.854 + 0.519i)14-s + (−0.917 + 0.398i)16-s + (0.816 − 0.576i)17-s + (−0.997 + 0.0682i)18-s + (−0.990 − 0.136i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.779 - 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.779 - 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(235\)    =    \(5 \cdot 47\)
Sign: $-0.779 - 0.626i$
Analytic conductor: \(1.09133\)
Root analytic conductor: \(1.09133\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{235} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 235,\ (0:\ ),\ -0.779 - 0.626i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2715691276 - 0.7712496550i\)
\(L(\frac12)\) \(\approx\) \(0.2715691276 - 0.7712496550i\)
\(L(1)\) \(\approx\) \(0.8175659204 - 0.4459108136i\)
\(L(1)\) \(\approx\) \(0.8175659204 - 0.4459108136i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
47 \( 1 \)
good2 \( 1 + (0.631 - 0.775i)T \)
3 \( 1 + (-0.398 + 0.917i)T \)
7 \( 1 + (-0.942 - 0.334i)T \)
11 \( 1 + (0.576 - 0.816i)T \)
13 \( 1 + (-0.997 - 0.0682i)T \)
17 \( 1 + (0.816 - 0.576i)T \)
19 \( 1 + (-0.990 - 0.136i)T \)
23 \( 1 + (-0.631 - 0.775i)T \)
29 \( 1 + (-0.0682 - 0.997i)T \)
31 \( 1 + (0.917 - 0.398i)T \)
37 \( 1 + (-0.519 + 0.854i)T \)
41 \( 1 + (-0.460 - 0.887i)T \)
43 \( 1 + (-0.979 + 0.203i)T \)
53 \( 1 + (0.887 - 0.460i)T \)
59 \( 1 + (-0.203 + 0.979i)T \)
61 \( 1 + (0.854 - 0.519i)T \)
67 \( 1 + (-0.942 + 0.334i)T \)
71 \( 1 + (-0.775 + 0.631i)T \)
73 \( 1 + (0.730 + 0.682i)T \)
79 \( 1 + (-0.962 - 0.269i)T \)
83 \( 1 + (0.816 + 0.576i)T \)
89 \( 1 + (0.990 - 0.136i)T \)
97 \( 1 + (0.398 - 0.917i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−26.18407490507419054469281484370, −25.36432516851750596657436039618, −24.898768386793222063332829219621, −23.72683080501111750622666976860, −23.13056855096335338376475216179, −22.26349400642832427169707967539, −21.53778964377855519619799890669, −19.91415068005541821601013468707, −19.14988154181424097731402880309, −17.94815941715858698446291373912, −17.107822540579004199081659673425, −16.424144215495226054187693735143, −15.14142848511116012074838890355, −14.317082033045836945802136905665, −13.17289410873786351594407438568, −12.36774969326150831238203205660, −11.92192054513522477558563173000, −10.11782745585136079225025993837, −8.80536804123542433011872149930, −7.61025777779121400684901593420, −6.770919709821614305610390660882, −5.98975240379373021277452503997, −4.87722776356625873611400266957, −3.41683286662805843576663490491, −2.0550173102328805862685218166, 0.47888370664226600537179724245, 2.66205367922221826347599916248, 3.6618881177557603590566099296, 4.58254760466284340953038367809, 5.78158439179568567128865302669, 6.645937469460822493441738104, 8.71035714635302437932982590596, 9.90231897603710343851653252377, 10.25837468612392134088022552251, 11.581023362272860544989878573063, 12.23066065765696385744974540472, 13.49571587112558648222965426326, 14.44282511840757221822238399569, 15.366883264476777114011609164600, 16.44351256310010063125848595109, 17.22547316729328876040780428795, 18.80985521720271248384611831932, 19.54633868624370467901781698529, 20.49916428521327564427230239675, 21.39030391574670822922441938851, 22.257366144068759536195673641336, 22.73525624573831192147241304089, 23.71976019167877255465722972606, 24.816695917132888281155703312320, 26.17031403748840642043495895252

Graph of the $Z$-function along the critical line